68 6. Brouwer’s fixed point theorem, separation, invariance of dimension
Then g|Sn−1 = idSn−1 , the identity on Sn−1.
Now let i : Sn−1 → Dn be the inclusion and consider
id = id = (g ◦ i) = g ◦ i ,
a map
n |
1 |
i |
n |
g |
n |
1 |
; Z/2). |
SHn−1(S |
− |
; Z/2) −→ SHn−1 |
(D |
; Z/2) −→ SHn−1(S |
− |
||
By Theorem 5.3 we have SHn−1(Sn−1; Z/2) = Z/2. |
Thus the identity on |
||||||
SHn−1(Sn−1; Z/2) is non-trivial. On the other hand, since Dn is homotopy
equivalent to a point, SH |
n−1 |
(Dn; Z/2) |
|
SH |
n−1 |
(pt; Z/2) = |
{ |
} |
, implying |
|
= |
|
0 |
|
a contradiction. q.e.d.
2. A separation theorem
As an application of the relation between the number of path components of a space X and the dimension of SH0(X; Z/2), we prove a theorem which generalizes a special case of the Jordan curve theorem (see, e.g., [Mu]). A topological manifold M is called closed if it is compact and has no boundary.
Theorem 6.2. Let M be a closed, path connected, topological manifold of dimension n − 1 and let f : M × (− , ) → U Rn be a homeomorphism onto an open subset U of Rn. Then Rn − f(M) has two path components.
In other words, a nicely embedded closed topological manifold M of dimension n − 1 in Rn separates Rn into two connected components. Here “nicely embedded” means that the embedding can be extended to an embedding of M × (− , ). If M is a smooth submanifold, then it is automatically nicely embedded [B-J].
Proof: Denote Rn |
− n |
|
1 |
|
f(M) by V . Since U |
|
V = Rn and SH (Rn; Z/2) |
= |
SH1(pt; Z/2) = 0 (R is contractible), the Mayer-Vietoris sequence gives the exact sequence
0 → SH0(U ∩V ; Z/2) → SH0(U; Z/2) SH0(V ; Z/2) → SH0(Rn; Z/2) → 0.
Now, U ∩V is homeomorphic to M ×(− , )−M ×{0} and thus has two path components. Then Theorem 4.6 implies SH0(U ∩ V ; Z/2) is 2-dimensional. The space U is homeomorphic to M × (− , ) which is path connected implying that the dimension of SH0(U) is 1. Since the alternating sum of the dimensions is 0 we conclude dimZ/2 SH0(V ; Z/2) = 2, which by Theorem
3. Invariance of dimension |
69 |
4.6 implies the statement of Theorem 6.2. q.e.d.
As announced earlier, although the result is equivalent to a statement about SH0(Rn−f(M); Z/2), the proof uses higher homology groups, namely the vanishing of SH1(Rn; Z/2).
3. Invariance of dimension
An m-dimensional topological manifold M is a space which is locally homeomorphic to an open subset of Rm. This is the case if and only if all points m M have an open neighbourhood U m which is homeomorphic to Rm. A fundamental question which arises immediately is whether the dimension of a topological manifold is a topological invariant. That is, could it be that M is both an m-dimensional manifold and an n-dimensional manifold for
n = m? In particular, are there n and m with n = m but with Rm Rm?
=
In this section we answer both these questions in the negative.
The key idea which we use is the local homology of a space. To define the local homology of a topological space X at a point x X, we consider the space X X−x C(X − x), the union of X and the cone over X − x, where
CY = Y × [0, 1]/Y ×{0} and we identify Y × {1} in CY with Y . Observe that X X−x C(X − x) = C(X) − (x × (0, 1)). We define the local homology
of X at x as SHk(X C(X − x); Z/2). We will use the local homology of a topological manifold to characterize its dimension. For this, we need the following consideration.
Lemma 6.3. Let M be a non-empty m-dimensional topological manifold. Then for each x M we have
SH |
(M |
|
C(M |
− |
x); Z/2) = |
|
Z/2 |
k = m |
k |
|
|
|
0 |
otherwise. |
Proof: Since M is non-empty, there is an x M and so we choose a homeomorphism ϕ from the open ball Bm to an open neighborhood of x.
We apply the Mayer-Vietoris sequence and decompose M C(M − x) into
U := C(M − x) and V := ϕ(Bm − {0}) × (12 , 1] {x}. The projection of V to ϕ(Bm) is a homotopy equivalence and so V is contractible. Also U is
contractible, since it is a cone. U ∩ V is homotopy equivalent (again via the projection) to ϕ(Bm − {0}) and so U ∩ V is homotopy equivalent to Sm−1. The reduced Mayer-Vietoris sequence is
·· · → SHk(U; Z/2) SHk(V ; Z/2) → SHk(M C(M − x); Z/2)
→SHk−1(U ∩ V ; Z/2) → · · · .
70 6. Brouwer’s fixed point theorem, separation, invariance of dimension
Since SHk(U; Z/2) and SHk(V ; Z/2) are zero and
SHk−1(Sm−1; Z/2), we have an isomorphism |
k−1 |
|
|||||||
k |
(M |
|
C(M |
− |
x); Z/2) |
|
SH |
(S |
|
SH |
|
|
= |
|
|||||
and the statement follows from 5.3. q.e.d.
SH |
k−1 |
(U |
∩ |
V ; Z/2) |
|
|
|
= |
m−1; Z/2)
Now we are in position to characterize the dimension of a non-empty topological manifold M in terms of its local homology. Namely by 6.3 we know that dim M = m if and only if SHm(M C(M − x); Z/2) = 0, where x is an arbitrary point in M. If f : M → N is a homeomorphism, then f can be extended to a homeomorphism g : M C(M −x) → N C(N −g(x)) and so the corresponding local homology groups are isomorphic. Thus
dim M = dim N.
We summarize our discussion with:
Theorem 6.4. Let f : M → N be a homeomorphism between non-empty manifolds. Then
dim M = dim N.
Remark: Let Y X be a subspace, then the reduced homology of X C(Y ) is called the relative homology of Y X and it is denoted by
SHk(X, Y ; Z/2) := SHk(X C(Y ); Z/2).
4.Exercises
(1)Give a formula for the map g : Dn → Sn−1 described in Theorem 6.1 and prove that it is continuous.
(2)Let A Mn(R) be a matrix whose entries are positive (non-negative). Show that A has a positive (non-negative) eigenvalue.
Chapter 7
Homology of some important spaces and the Euler characteristic
1. The fundamental class
Given a space X it is very useful to have some explicit non-trivial homology classes. The most important example is the fundamental class of a compact m-dimensional Z/2-oriented regular stratifold S which we introduced as [S]Z/2 := [S, id] SHm(S; Z/2). We have shown that for a sphere the fundamental class is non-trivial. In the following result, we generalize this.
Proposition 7.1. Let S be a compact m-dimensional Z/2-oriented regular stratifold with Sm = . Then the fundamental class [S]Z/2 SHm(S; Z/2) is non-trivial.
Proof: The 0-dimensional case is clear and so we assume that m > 0. We reduce the proof of the statement to the special case of spheres where it is already known. For this we consider a smooth embedding ψ : Bm → Sm, where Bm is the open unit ball, and we decompose S as ψ(Bm) =: U and S − ψ(0) =: V . Then U ∩ V = ψ(Bm − 0). We want to determine d([S]Z/2), where d is the boundary operator in the Mayer-Vietoris sequence corresponding to the covering of S by U and V . We choose a smooth function η : [0, 1] → [0, 1], which is 0 near 0, 1 near 1 and η(t) = t near 1/2, and then define ρ : S → [0, 1] by mapping ψ(x) to η(||x||) and S − im ψ to 1. Then
71